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| ==Algemene informatie==
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| ==Informatie over het examen==
| | =Informatie over het examen= |
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| ==De afgelopen examens== | | =Examens= |
| === Maandag 14 januari 2013 === | | == 2011-2012 == |
| [[Media:2013-01-14_examen.png]]
| | [[Media:16 januari 2012.pdf|16 januari 2012 (NM)]] |
| === Woensdag 1 februari 2012 ===
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| [[Media:Symmetries.jpeg]] | |
| === Maandag 23 januari 2012 ===
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| | [[Media:20 januari 2012.pdf|20 januari 2012 (NM)]] |
| ## Explain the relation between symmetry and conservation laws in quantum mechanics.
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| ## If turning on a perturbation has the effect of splitting the energy levels, then what can you generally infer about the system before the perturbation was turned on?
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| # Let <math>|njm></math> be the energy eigenstates of a particle in some spherically symmetrical potential, with j,m the usual angular momentum quantum numbers. A perturbation <math>H \rightarrow H + e(t)W</math> is applied, which will cause transitions between these states. Give as many selection rules as you can for the first order transition matrix element <math><njm|W|njm></math>, when (i) <math>W = e^{-r^2}</math>, (ii) <math>W = (x^2 - y^2)e^{-r^2}</math> , (iii) <math>W=L_x</math>.
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| # A system consists of 100 spin 2 particles. Construct a state with total spin quantum numbers (J,M)=(200,199).
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| # Just by counting degeneracies, find the total angular momentum of the ground state of eight non-interacting identical spin 1/2 particles in a 3D harmonic oscillator potential. Take into account the Pauli exclusion principle.
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| # A system with rotational and time reversal invariance whose energy levels are nondegenerate, except for the degeneracies implied by rotational invariance cannot have a permanent electric dipole moment in any energy eigenstate. Show this by combining time reversal invariance with the Wigner-Eckart theorem (to relate dipole and Angular momentum matrix elements).
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| === Vrijdag 20 januari 2012 ===
| | [[Media:23 januari 2012.pdf|23 januari 2012 (NM)]] |
| # The wigner eckart theorem: what's it good for? (in a few lines)
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| # Under what circumstances is it possible to simultaneously measure with arbitrary precision energy and (i) position (ii) momentum (iii) angular momentum and (iv) velocity, of particle in a static electric field? And in a magnetic field?
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| # A particle is in a state with wave function <math>\psi(x,y,z)=(x+y)f(r)</math> where <math>r=\sqrt{x^2+y^2+z^2}</math>.
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| ## argue using simple symmetry reasoning that <math><\chi|\psi>=0</math> for any <math>|\chi></math> that is even under parity or odd under the exchange of x and y coordinates.
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| ## what are the possible outcomes for measurements of <math>L^2</math> and <math>L_{z}</math>, and with what probabilities?
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| ## let |nlm> be hydrogen wave function. For which values of l,m is it guaranteed that <math><nlm|z|\psi>=0</math>? Same for <math><nlm|x^2-y^2|\psi></math> and <math><nlm|L+|\psi></math>.
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| # A system consists out of 100 spin 1 particles. Can you explicitly construct a state with total spin quantum number |J=100,m=99>?
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| ## two atoms are slowly brought towards each other. what will happen to their energy spectra?
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| ## three helium 3 atoms walk into a bar. all energy levels of this trio are double degenerate. what can be done to undo this doubling?
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| === Maandag 16 januari 2012 === | | [[Media:1 februari 2012.pdf|1 februari 2012 (NM)]] |
| # symmetries in quantum mechanics: what are they good for?
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| # Say we have an orthonormal set of three states {psi_x, psi_y, psi_x} (say of some atom), on which rotations act in the standard vector representation, i.e. they transform among each other in the same way as a vector (x,y,z) in R³
| | == 2012-2013 == |
| ## How do the operators corresponding to angular momentum Lx, Ly and Lz act on these states?
| | [[Media:Examen Symmetries in QM (2012-2013)(januari).pdf|14 januari 2013 (NM)]] |
| ## What are the possible values of Lz?
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| ## if we prepare a beam of these atoms, prepared to be in states restricted to be linear combinations of the above three states, and we first pass this beam through a filter that allows through only atoms with maximally positive spin in the z-direction, next through a filter that allows only atoms with maximally positive spin in the x-direction, and last through a filter that allows only atoms with maximally negative spin in the z-direction, then what fraction of the original beam will survive?
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| ## Can this result change if the beam passes through some homogeneous magnetic field between two subsequent detectors?
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| ## A spin 1 particle at rest decays into a spin 1/2 particle B and a spin 1/2 particle C. What are the possible values of the z-components of spin B and C assuming the final (center of mass) orbital angular momentum is measured to be zero?
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| ## What are the other possible values of the final orbital angular momentum that could be measured, and what are the corresponding values of the spins of B and C?
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| ## How do the possibilities get further reduced if you know the intrinsic parities of A, B and C are all even?
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| ## Restricting again to the zero orbital angular momentum sector, and assuming all initial spin states are equally likely, what are the probabilities for the z-components of the spin of B and C?
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| ## How would you solve this last problem for a spin 3 particle A decaying into a spin 1 particle B and a spin 2 particle C? (optionally: solve it)
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| [[Categorie:mf]] | | [[Categorie:mf]] |